Proof. 
We follow the idea included in [1]}, Lemma 3).
Fix a sequence \( a_n\)  of positive numbers such that \( \sum _{n=1}^{\infty } a_n =1.\)  By the proof of Theorem REF  and Lemma REF  the mapping \( Q_n: X \rightarrow F_n\)  defined by \( Q_nx = x - f_n(x-z_n)y_n\)  if \( f_n(x) > d_n\)  and
\(Q_nx =x\)  in the opposite case, is a contractive projection from \(X\)  onto \(F_n.\)  (Here for each \( n \in \mathbb {N}\)  \(z_n\)  is so chosen that \( f_n(z_n)=d_n.)\)  Define \( Q: X \rightarrow X\)  by
\(Qx = \sum _{n=1}^{\infty } a_n Q_n .\) 
